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Characterization of generalized Petersen graphs that are Kronecker covers Matjaž Krnc, Tomaž Pisanski To cite this version: Matjaž Krnc, Tomaž Pisanski. Characterization of generalized Petersen graphs that are Kronecker covers. 2018. hal-01804033v1 HAL Id: hal-01804033 https://hal.archives-ouvertes.fr/hal-01804033v1 Preprint submitted on 31 May 2018 (v1), last revised 3 Nov 2019 (v6) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Characterization of generalized Petersen graphs that are Kronecker covers∗ Matjaž Krncy and Tomaž Pisanskiz FAMNIT, University of Primorska, Slovenia May 31, 2018 Abstract The family of generalized Petersen graphs G (n; k), introduced by Coxeter et al. [4] and named by Mark Watkins (1969), is a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. The Kronecker cover KC (G) of a simple undirected graph G is a a special type of bipartite covering graph of G, isomorphic to the direct (tensor) product of G and K2. We characterize all the members of generalized Petersen graphs that are Kronecker covers, and describe the structure of their respective quotients. We observe that some of such quotients are again generalized Petersen graphs, and describe all such pairs. Keywords— Generalized Petersen graphs, Kronecker cover MSC: 57M10, 05C10, 05C25 1 Introduction The generalized Petersen graphs, introduced by Coxeter et al. [4] and named by Watkins [15], form a very interesting family of trivalent graphs that can be described by only two integer parameters. They include Hamiltonian and non-Hamiltonian graphs, bipartite and non-bipartite graphs, vertex-transitive and non-vertex-transitive graphs, Cayley and non- Cayley graphs, arc-transitive graphs and non-arc transitive graphs, graphs of girth 3; 4; 5; 6; 7 or 8. Their generalization to I-graphs does not introduce any new vertex-transitive graphs but it contains also non-connected graphs and has in special cases unexpected symmetries. n Following the notation of Watkins [15], for a given integers n and k < 2 , we can define a generalized Petersen graph G (n; k) as a graph on vertex-set fu0; : : : ; un−1; v0; : : : ; vn−1g. The edge-set may be naturally partitioned into three equal parts (note that all subscripts ∗Dedicated to Mark Watkins on the Occasion of his 80th Birthday. [email protected] [email protected] 1 n−1 are assumed modulo n): the edges EO (n; k) = fuiui+1gi=0 from the outer rim, inducing a n−1 cycle of length n; the edges EI (n; k) = fvivi+kgi=0 from the inner rims, inducing gcd(n; k) n n−1 cycles of length gcd(n;k) ; and the edges ES (n; k) = fuivigi=0 , also called spokes, that induce a perfect matching in G (n; k). Hence the edge-set may be defined as E (G (n; k)) = EO (n; k)[ EI (n; k) [ ES (n; k). Various aspects of the structure of the mentioned family has been observed. Examples include identifying generalized Petersen graphs that are Hamiltonian [1] or Cayley [13, 10], or finding their automorphism group [14, 11, 7]. Also, a related generalization to I-graphs has been introduced in the Foster census [3], and further studied by Boben et al. [2]. Theory of covering graphs became one of the most important and successful tools of algebraic graph theory. It is a discrete analog of the well known theory of covering spaces in algebraic topology. In general, covers depend on the values called voltages assigned to the edges of the graphs. Only in some cases the covering is determined by the graph itself. One of such cases is the recently studied clone cover [9]. The other, more widely known case is the Kronecker cover. The Kronecker cover KC (G) (also called bipartite or canonical double cover) of a sim- ple undirected graph G is a bipartite covering graph with twice as many vertices as G. Formally, KC (G) is defined as a tensor product G × K2, i.e. a graph on a vertex-set 0 00 0 00 00 0 V (KC (G)) = fv ; v gv2V (G), and an edge-set E (KC (G)) = fu v ; u v guv2E(G). Some re- cent work on Kronecker covers includes Gévay and Pisanski [6] and Imrich and Pisanski [8]. In this paper, we study the family of generalized Petersen graphs in conjunction with the Kronecker cover operation. Namely, in the next section we state our main theorem characterizing all the members of generalized Petersen graphs that are Kronecker covers, and describing the structure of their corresponding quotient graphs. In Section 3 we focus on the necessary and sufficient conditions for a generalized Petersen graph to be a Kronecker cover while in Section 4 we complement the existence results with the description of the structure of the corresponding quotient graphs. We conclude the paper with some remarks and directions for a possible future research. 2 Main result In order to state the main result we need to introduce the graph and two 2-parametric families of cubic, connected graphs. Let H be a graph defined by the following procedure: take the Cartesian product K3P3, remove the edges of the middle triangle, add a new vertex in the middle and connect it to all three 2-vertices. Note that the graph H is mentioned in the paper [8] where it is depicted in Figure 1. The Desargues graph G (10; 3) ' KC (H) may be similarly defined by first taking K6P3, removing the edges of the middle hexagon, adding two new vertices in the middle and alter- nately connecting them to the consecutuve vertices of the middle hexagon. The mentioned construction of H and the standard drawing of Desargues graph are depicted on Figure 1. To describe the quotients of generalized Petersen graphs, we use the LCF notation, named by developers Lederberg, Coxeter and Frucht, for the representation of cubic hamiltonian 2 Figure 1: The Desargues graph and its both quotients; H and the Petersen graph. graphs (for extended description see [12]). In a Hamiltonian cubic graph, the vertices can be arranged in a cycle, which accounts for two edges per vertex. The third edge from each vertex can then be described by how many positions clockwise (positive) or counter-clockwise (negative) it leads. The basic form of the LCF notation is just the sequence [a0; a1; : : : ; an−1] of numbers of positions, starting from an arbitrarily chosen ver- tex and written in square brackets. To state our results, we only use a special type of such LCF-representable graphs, namely C+(n; k) and C−(n; k), which we define below. Definition 1. Assuming all numbers are modulo n, define graphs hn n n n i C+(n; k) = ; + (k − 1); + 2(k − 1);:::; + (n − 1)(k − 1) ; 2 2 2 2 and similarly hn n n n i C−(n; k) = ; − (k + 1); − 2(k + 1);:::; − (n − 1)(k + 1) : 2 2 2 2 In [8] it was proven that G (10; 3) is Kronecker cover of two non-isomorphic graphs. Here we prove among other things that this is the only generalized Petersen graph that is a multiple Kronecker cover. Every other generalized Petersen graph is either a Kronecker cover of a single graph or it is not a Kronecker cover at all. More precisely; Theorem 1. Among the members of the family of generalized Petersen graphs, G (10; 3) is the only graph that is the Kronecker cover of two non-isomorphic graphs, the Petersen graph and the graph H. For any other G ' G (n; k), the following holds: a) If n ≡ 2 (mod 4) and k is odd, G is a Kronecker cover. In particular n a1) if 4k < n, the corresponding quotient graph is G 2 ; k , and 3 n n a2) if n < 4k < 2n the quotient graph is G 2 ; 2 − k . k2−1 b) If n ≡ 0 (mod 4) and k is odd, G is a Kronecker cover if and only if n j 2 and n k < 2 . Moreover, + b1) if k = 4t + 1 the corresponding quotient is C (n; k) while − b2) if k = 4t + 3 the quotient is C (n; k). c) Any other generalized Petersen graph is not a Kronecker cover. For k = 1 and even n each G(n; 1) is a Kronecker cover. However, if n = 4t case b1) applies and the quotient graph is even Moebius ladder. For G(4; 1) the quotient is K4 = M4. Similarly, the 8-sided prism G(8; 1) is a Kronecker cover of M8. In case n = 4t + 2 the case a1) applies and the quotient is G(n=2; 1). For instance, the 6-sided prism is a Kronecker cover of a 3-sided prism. For k > 1 the smallest cases stated in Theorem 1 are presented in Table 1. It is well-known that any automorphism of a connected bipartite graph either preserves the two sets of bipartition or interchanges the two sets of bipartition. In the former case we call the automorphism colour preserving and in the latter case colour reversing. Clearly, the product of two color-reversing automorphisms is a color preserving automorphism and the collection of color preserving automorphisms determine a subgroup of the full automorphism group that is of index 2.

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